an:05552748
Zbl 1171.65014
Towers, John D.
Finite difference methods for approximating Heaviside functions
EN
J. Comput. Phys. 228, No. 9, 3478-3489 (2009).
00248659
2008
j
65D32 41A55 41A25 41A63
Heaviside function; level set method; quadrature; irregular region; singular source term; finite difference; regular grid; convergence rate; one-step algorithm; two-step algorithm; numerical examples
The paper presents two algorithms to evaluate the integral \(\mathcal{I}:=\int_\Omega f(\vec{x})d\vec{x}\), where \(\vec{x}\in \mathbb{R}^n\), \(\Omega=\{\vec{x}:\,u(\vec{x})>0\}\) and \(\partial\Omega=\{\vec{x}:\,u(\vec{x})=0\}\) is a compact manifold of codimension one. The method to be employed is motivated by the expression \(\mathcal{I}=\int_{\mathbb{R}^n} H(u(\vec{x}))f(\vec{x})d\vec{x}\), where \(H\) is the Heaviside function. The approach consists of approximating \(H\) by finite differencing its first few primitives, a technique already used by the author to approximate delta functions.
A brief presentation of the algorithms is given below. Let
\[
I(z)=\int_0^z H(\zeta)d\zeta\quad\text{and}\quad J(z)=\int_0^z I(\zeta)d\zeta.
\]
The following relationships are derived:
\[
I(u)=\langle\nabla J(u),\nabla u \rangle /|\nabla u|^2,\quad H(u)=\langle\nabla I(u),\nabla u \rangle /|\nabla u|^2,
\]
where \(\langle \cdot,\cdot \rangle\) stands for the inner product. By discretizing \(H(u)\) the one-step algorithm \(FDMH_1\) is obtained which converges at a rate of \(\mathcal{O}(h^2)\) when \(u\) is smooth enough. By discretizing both relationships the two-step algorithm \(FDMH_2\) is derived which can converge at a rate of \(\mathcal{O}(h^3)\). These results are validated by means of some numerical examples.
Jesus Ill??n Gonz??lez (Vigo)